Lectures on Superstring and M Theory Dualities
John H. Schwarz
TL;DR
The notes survey a unified web of dualities linking the five perturbative 10d superstring theories via S, T, and U dualities, and introduce M theory as a common nonperturbative framework. Central to the story are p-branes and BPS states, whose tensions and charges encode the nonperturbative spectrum and underpin cross-theory identifications, such as IIB on circles matching M-theory on tori and heterotic strings realizing brane-bound states. By tracing compactifications to 6d and 9d, Schwarz demonstrates how dualities organize massless spectra, anomaly cancellation, and moduli spaces (e.g., M_{5,5}, M_{4,n}, and K3-related structures), leading to a network where perturbative and nonperturbative sectors swap roles under S- and T-dualities. The results imply that what counts as a fundamental object in one description may appear as a solitonic or wrapped brane in another, and they reveal a rich 'duality of dualities' connecting heterotic, type I, type II, and M-theory in diverse dimensions. These insights have profound implications for understanding the landscape of consistent vacua, chirality, and potential nonperturbative phase structures in string theory.
Abstract
These lectures begin by reviewing the evidence for S duality of the toroidally compactified heterotic string in 4d that was obtained in the period 1992--94. Next they review recently discovered dualities that relate all five of the 10d superstring theories and a quantum extension of 11d supergravity called M theory. The study of p-branes of various dimensions (some of which are D-branes) plays a central role. The final sections survey supersymmetric string vacua in 6d and some of the dual constructions by which they can be obtained. Special emphasis is given to a class of N=1 models that exhibit ``heterotic-heterotic duality.''